Class 3: Housing Concepts, Household Bids PAI786: Urban Policy Class 3: Housing Concepts, Household Bids

Urban Policy: Housing Concepts Household Bids Outline of Class Land concepts Housing concepts Housing bids and locational equilibrium

Urban Policy: Housing Concepts Household Bids Some Perspective We are going to focus on concepts today (and in the next few classes)—not on policy. But these concepts help us to think more clearly about urban housing markets, And to think more clearly about urban policy when we focus on it a few classes from now.

Urban Policy: Housing Concepts Household Bids Land Concepts Land rent is the price for using one unit of land, say an acre, for one unit of time, say a year. Land value is the price of buying one unit of land, again say an acre. Land is an asset; like any asset its price (=value) is the present value of the benefits (=net rent) from owning it.

Urban Policy: Housing Concepts Household Bids Land Concepts, 2 Value of a Land Parcel Net Rent of Land in Year 2 Discount Rate Assumes RL is constant over time

Urban Policy: Housing Concepts Household Bids The Determination of Land Rent Land is an input; the price of land (= annual rent) is a derived demand—derived from its role in producing an output, say Q, which sells for PQ. In equilibrium, the price of an input equals the value of its marginal product, so:

Urban Policy: Housing Concepts Household Bids Land Rent, 2 One important lesson: The price of land is high in some locations because people are willing to pay a lot for housing or commercial activities at that location It is not correct to say that the price of housing is high because land is expensive.

Urban Policy: Housing Concepts Household Bids Land Rent, 3 Now consider this highly simplified example: Q must be shipped to a market The distance to the market, designated u, varies across firms. It costs $s to ship a unit of Q one mile. The marginal product of land equals a. Then land rent is determined by: Net Price of Product Marginal Product of Land

Urban Policy: Housing Concepts Household Bids Land Rent, 4 With a constant a, land rent is a linear function of distance, u. But when the price of land goes down, firms are likely to substitute land for capital. This increase in the amount of land at greater distances from the market leads to a lower MPL at greater distances And hence to a land-rent function with a slope that gets flatter as distance from the market increases.

Urban Policy: Housing Concepts Household Bids Land Rent and Distance from the Market without with substitution substitution < Figure 1 > R(u) A R(u) B Market u Market u

Urban Policy: Housing Concepts Household Bids Housing is measured in units of housing services = H H = quality-adjusted square feet. Depends on housing characteristics (X1, X2, …). P = the price per unit of H per year. R = rent for a housing unit = PH. If the unit is an apartment, R = contract rent. If the unit is owner-occupied, R is not observed.

Urban Policy: Housing Concepts Household Bids The price, P, of housing, H, depends on many locational traits. Let’s start with distance to a worksite, u. More later! People will pay more per unit of H if they have good access to a worksite than if they do not. So P = P{u}.

Urban Policy: Housing Concepts Household Bids Housing Concepts, Continued V = the value of a housing unit = the present value of the rental flow (not observed for renters). So, with a long lifetime, T, for housing:

Urban Policy: Housing Concepts Household Bids How Does a CBD Worker Decide Where To Live? She compares the marginal benefit (MB) and the marginal cost (MC) of moving one mile farther from the CBD. Drop in Price per Quality-Adjusted Square Foot Increase in Commuting Distance

Urban Policy: Housing Concepts Household Bids How Does a CBD Worker Decide Where To Live? (Continued) She then keeps moving out until she comes to the location (u*) at which MB equals MC: Lower Housing Cost Higher Commuting Cost

Urban Policy: Housing Concepts Household Bids Tradeoff Between Housing and Commuting Costs Note: This MB curve is just hypothetical; we do not (yet) know its shape.

Urban Policy: Housing Concepts Household Bids The Twist: How Housing Prices Are Determined Now suppose that all households are alike (an assumption to be relaxed!). Then they all pick the same u*! This is impossible, so P{u} adjusts until people are equally satisfied no matter where they live. This is called locational equilibrium.

Urban Policy: Housing Concepts Household Bids The Twist: How Housing Prices Are Determined, 2 In other words, P{u} adjusts until MB=MC at all locations, or that is, until the slope of the P{u} function, ΔP{u}/Δu, equals –t/H, the transportation cost/housing trade-off.

Urban Policy: Housing Concepts Household Bids The Bid Function for Housing (Price per Unit of Housing Services) Slope = ΔP/Δu = -t/H < Figure 3 > P(u) CBD u ΔP Δu

Urban Policy: Housing Concepts Household Bids The Twist: How Housing Prices Are Determined, 3 Because the slope is negative, P{u} is higher closer to the CBD than it is in the suburbs. When P{u} is high, people substitute away from housing so that H is low. When H is low, the slope of P{u}, namely, -t/H, is high in absolute value. It follows that P{u} is steep near the city center but flattens as one moves out toward the suburbs.

Urban Policy: Housing Concepts Household Bids The Twist: How Housing Prices Are Determined, 4 This relationship between housing prices and access to jobs is not just hypothetical. Many studies have found that, controlling for other things, the price per quality-adjusted square foot is lower at greater distances from a worksite. The following figure gives an example from a study of mine on the Cleveland area; the price per unit of H is 35% higher next to the center than at the outer edge.

Urban Policy: Housing Concepts Household Bids

Urban Policy: Housing Concepts Household Bids Finding the Edge of the City Urban activities must compete with rural activities for access to land. Suppose P* is the opportunity cost of pulling resources out of agriculture and into housing. Then urban activities will take place out to the point, say, u*, at which the price of housing exceeds P*.

Urban Policy: Housing Concepts Household Bids Determining the Outer Edge of the Urban Area < Figure 3A > P(u) P* CBD u* u

Urban Policy: Housing Concepts Household Bids Policy Questions and Bid Functions Some policies affect a single urban area. If they make the area more attractive, people move in; otherwise, people move out to other areas. These policies are analyzed with an “open” model. Other policies affect all urban areas. These policies do not give anyone an incentive to move out of an area. These policies are analyzed with a “closed” model.

Urban Policy: Housing Concepts Household Bids The Height of the Bid Function and the Size of the Area To understand the distinction between open and closed models, recall that we derived a formula for the slope of P{u}, not for its height. As the height of P{u} goes up, The level of satisfaction in an urban area goes down, And the population goes up.

Urban Policy: Housing Concepts Household Bids The Height of the Bid Function and the Size of the Urban Area < Figure 3B > P(u) CBD u Lower Utility

Urban Policy: Housing Concepts Household Bids Open versus Closed Models In an open model, one selects the height of P{u} that yields the same level of satisfaction as a household can obtain in another urban area. At any other height, people would move in or out. In a closed model, one selects the height of P{u} that makes the area large enough to fit all its population.

Urban Policy: Housing Concepts Household Bids Open versus Closed Examples The height of the bid function is like the cost of living. If the height is higher, housing costs more and income does not go as far. If the heights of the bid functions are different in two cities in an open model, incomes must be higher (or amenities better) in the location with the higher prices—or else nobody would live there!

Urban Policy: Housing Concepts Household Bids Open versus Closed Examples, 2 Suppose one city in a region cleans up its air and no other city does. The impacts are given by an open model. People move in and housing prices go up until the higher cost of living offsets the utility gain from cleaner air! Suppose all cities in the region clean their air. The impacts are given by a closed model. Nobody has an incentive to move out and utility goes up due to cleaner air—but of course everyone is better off.